Part II : Mathematics
Section-I (Single Correct Choice Type)
This Section contains 8 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.
1. If the angles A, B and C of a triangle are in an arithematic progression and if a, b and c denote the lengths of sides opposite to A, B and C respectively, then the value of the expression 
is
(A) 1/2
(B) √3/2
(C) 1
(D) √3
2. Equation of the plane containing the straight line 
and perpendicular to the plane containing the straight lines 
is
(A) x + 2y – 2z = 0
(B) 3x + 2y – 2z = 0
(C) x – 2y + z = 0
(D) 5x + 2y – 4z = 0
3. Let ω be a complex cube root of unity with ω ≠ A fair die is thrown three times. If r1, r2 and r3 are the numbers obtained on the die, then the probability that 
(A) 1/18
(B) 1/9
(C) 2/9
(D) 1/36
4. Let P, Q, R and S be the points on the plane with position vectors 
The quadrilateral PQRS must be a
(A) parallelogram, which is neither a rhombus nor a rectangle
(B) square
(C) rectangle, but not a square
(D) rhombus, but not a square
5. The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system 
has exactly two distinct solutions, is
(A) 0
(B) 29 – 1
(C) 168
(D) 2
6. The value of 
is
(A) 0
(B) 1/12
(C) 1/24
(D) 1/64
7. Let p and q be real numbers such that p ≠ 0, p3 ≠ q and p3 ≠ − If α and β are nonzero complex numbers satisfying α + β = −p and α3 + β3 = q, then a quadratic equation having 
as its roots is
(A) (p3 + q)x2 − (p3 + 2q)x + (p3 + q) = 0
(B) (p3 + q)x2 − (p3 − 2q)x + (p3 + q) = 0
(C) (p3 − q)x2 − (5p3 − 2q)x + (p3 − q) = 0
(D) (p3 − q)x2 − (5p3 + 2q)x + (p3 − q) = 0
8. Let f, g and h be real-valued functions defined on the interval [0, 1] by 
and 
. If a, b and denote, respectively, the absolute maximum of f, g and h on [0, 1], then
(A) a = b and c ≠ b
(B) a = c and a ≠ b
(C) a ≠ b and c ≠ b
(D) a = b = c
Section − II (Multiple Correct Choice Type)
This section contains 5 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONE OR MORE may be correct.
9. Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches a circle or radius r having AB as its diameter, then the slope of the line joining A and B can be
(A) −1/r
(B) 1/r
(C) 2/r
(D) −2/r
10. Let ABC be a triangle such that 
and let a, b and c denote the lengths of the sides opposite to A, B and C respectively. The value(s) of for which a = x2 + x + 1, b = x2 – 1 and c = 2x + 1 is (are)
(A) −(2 + √3)
(B) 1 + √3
(C) 2 + √3
(D) 4√3
11. Let z1 and z2 be two distinct complex numbers and let z = (1 – t)z1 + tz2 for some real number t with 0 < t < 1. If Arg(w) denotes the principal argument of a nonzero complex number w, then
(A) |z – z1| + |z – z2| = |z1 – z2|
(B) Arg(z – z1) = Arg(z – z2)
(C) 
(D) Arg(z – z1) = Arg(z2 – z1)
12. Let f be a real-valued function defined on the interval (0, ∞) by 
Then which of the following statement(s) is (are) true?
(A) fʹʹ(x) exists for all x ∈ (0, ∞)
(B) fʹ(x) exists for all x ∈ (0, ∞) and fʹ is continuous on (0, ∞), but not differentiable on (0, ∞)
(C) there exists α > 1 such that |fʹ(x)| < |f(x)| for all x ∈ (α, ∞)
(D) there exists β > 0 such that |f(x)| + |fʹ(x)| ≤ β for all x ∈ (0, ∞)
13. The value(s) of 
is (are)
(A) 
(B) 
(C) 
(D) 
SECTION − III (Paragraph Type)
This section contains 2 paragraphs. Based upon the first paragraph 2 multiple choice questions and based upon the second paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices A), B), C) and D) out of WHICH ONLY ONE CORRECT.
Paragraph for Questions 14 to 16
Let p be an odd prime number and Tp be the following set of 2 × 2 matrices :
14. The number of A in Tp such that A is either symmetric or skew-symmetric or both, and det (A) divisible by p is
(A) (p – 1)2
(B) 2(p – 1)
(C) (p – 1)2 + 1
(D) 2p – 1
15. The number of A in Tp such that the trace of A is not divisible by p but det(A) is divisible by p is
[Note : The trace of a matrix is the sum of its diagonal entries.]
(A) (p – 1) (p2 – p + 1)
(B) p3 – (p – 1)2
(C) (p – 1)2
(D) (p – 1) (p2 – 2)
16. The number of A in Tp such that det(A) is not divisible by p is
(A) 2p2
(B) p3 – 5p
(C) p3 – 3p
(D) p3 – p2
Paragraph for Questions 17 to 18
The circle x2 + y2 – 8x = 0 and hyperbola 
intersect at the points A and B.
17. Equation of a common tangent with positive slope to the circle as well as to the hyperbola is
(A) 2x – √5y – 20 = 0
(B) 2x – √5 + 4 = 0
(C) 3x – 4y + 8 = 0
(D) 4x – 3y + 4 = 0
18. Equation of the circle with AB as its diameter is
(A) x2 + y2 – 12x + 24 = 0
(B) x2 + y2 + 12x + 24 = 0
(C) x2 + y2 + 24x – 12 = 0
(D) x2 + y2 – 24x – 12 = 0
SECTION − IV (Integer Type)
This section contains TEN questions. The answer to each question is a single digit integer ranging from 0 to 9.
19. The number of values of θ in the interval 
such that 
for n = 0, ±1, ±2 and tan θ = cot 5θ as well as sin 2θ = cos 4θ is
20. The maximum value of expression 
is
21. If 
are vectors in space given by 
then the value of 
is
22. The line 2x + y = 1 is tangent to the hyperbola 
If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is
23. If the distance between the plane Ax – 2y + z = d and the plane containing the lines 
and 
is √6, then |d| is
24. For any real number x, let [x] denote the largest integer less than or equal to x. Let f be a real valued function defined on the interval [−10, 10] by
Then the value of 
25. Let ω be the complex number 
Then the number of distinct complex numbers z satisfying 
is equal to
26. Let Sk, k = 1, 2,…., 100, denote the sum of the infinite geometric series whose first term is 
and the common ratio is 1/k. Then the value of 
is
27. The number of all possible of θ, where 0 < θ < π, for which the system of equations
(y + z) cos 3θ = (xyz) sin 3θ.
(xyz) sin 3θ = (y + 2z) cos 3θ + y sin 3θ
have a solution (x0, y0, z0) with y0z0 ≠ 0, is
28. Let f be a real-valued differentiable function on R (the set of all real numbers) such that f[1] = 1. If the y-intercept of the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P, then the value of f(−3) is equal to
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